Cohomologie de Chevalley des graphes vectoriels

Abstract

The space T poly (R d ) of all tensor fields on R d , equipped with the Schouten bracket is a Lie algebra. The subspace of ascending tensors is a Lie subalgebra of T poly (R d ).In this paper, we compute the cohomology of the adjoint representations of this algebra (in itself and T poly (R d )), when we restrict ourselves to cochains defined by aerial Kontsevitch's graphs like in our previous work (Pacific J of Math, vol 229, no 2, (2007) 257-292). As in the vectorial graphs case, the cohomology is freely generated by all the products of odd wheels. Résumé.L'espace des tenseurs ascendants est une sous algèbre de Lie de l'algèbre de Lie (graduée) T poly (R d ) des champs de tenseurs sur R d muni du crochet de Schouten.Dans cet article, on calcule la cohomologie des représentations adjointes de cette sous algèbre de Lie, en se restreignant à des cochaînes définies par des graphes de Kontsevich aériens comme dans [AAC1] et [AAC2]. On retrouve un résultat analogue à celui de la cohomologie des graphes vectoriels et linéaires: elle est librement engendrée par des produits de roues de longueurs impaires.

Chevalley cohomology for aerial Kontsevich graphs

“…With this shift of degree, T poly (R d ) becomes a Lie{1} algebra, and the Chevalley cohomology operator becomes ∂, acting on symmetrized mappings as follows (see [AAC1]): Proposition 2.1 (Symmetrized cohomology). If C is a n-linear, | |-symmetric mapping, then its coboundary ∂C is…”

“…Then ∆ is either a wheel or a line. These graphs and their cohomology were studied, for instance, in [AAC1].…”

“…In [AAC1], it is proved that the cohomology of lines vanishes and the cohomology of wheels is the space generated by the products W 2k1+1 ∧ W 2k2+1 ∧ · · · ∧ W 2kp+1 , with 0 < k 1 < · · · < k p . As above, the cohomology is freely generated by the product of odd wheels W 2p+1 .…”

“…In [AAC1] and [AAC2] we considered only small subalgebras of T poly (R d ), where "small"means that we restricted attention to particular classes of graphs ∆.…”

“…We then compute the corresponding cohomology completely. As in [AAC1,AAC2], the cohomology is freely generated by odd wheels products.…”